Alex Elzenaar

Publications and preprints

1. Research announcement: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "On Thin Heckoid and Generalised Triangle Groups in \( \mathrm{PSL}(2,\mathbb{C}) \)", 2024. arXiv: 2409.04438 [math.GR].
2. Preprint: Alex Elzenaar, Jianhua Gong, Gaven Martin, and Jeroen Schillewaert. "Bounding deformation spaces of 2-generator Kleinian groups", 2024. arXiv:2405.15970 [math.CV].
3. Preprint: Alex Elzenaar and Shayne Waldron. "Putatively optimal projective spherical designs with little apparent symmetry", 2024. arXiv:2405.19353 [math.CO].
   Associated dataset: A.E. and S.W. A repository of spherical \( (t,t) \)-designs (Version 2). Zenodo, 2022. DOI: 10.5281/zenodo.8126277
4. Proceedings article: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds". In: 2021-22 MATRIX annals. Ed. by David R. Wood, Jan de Gier, Cheryl E. Prager, and Terrence Tao. MATRIX Book Series 5. Springer, 2024, pp. 31–74. DOI: 10.1007/978-3-031-47417-0_2. Preprint version: arXiv:2204.11422 [math.GT]. Corrected preprint: PDF.
5. Journal article: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "The combinatorics of the Farey words and their traces." In: Groups, Geometry, and Dynamics. Accepted, to appear. Preprint version: arXiv:2204.08076 [math.GT].
6. Journal article: Alex Elzenaar, Gaven Martin, and Jeroen Schillewaert. "Approximations of the Riley slice." In: Expositiones Mathematicae 41 (2023), pp. 20–54. DOI: 10.1016/j.exmath.2022.12.002. MR4557273. Preprint version: arXiv:2111.03230 [math.GT]. Corrected preprint: PDF.

Expository writing (unpublished)

1. The lake where they had hidden the reflections: Quasi-Fuchsian groups and their embeddings [This was a talk in a reading group.]
2. Knot knotes
3. Apocrypha and ephemera on the boundaries of moduli space [More detailed notes on deformations of geometrically finite Kleinian groups are under preparation.]
4. Uniformisation, equivariance, and vanishing: three kinds of functions hanging around your Riemann surface

Selected talks

Here is material (e.g. lecture notes, slides) from selected research talks I have given.

1. (Upcoming) December 2024: Deformations of 3-orbifold holonomy groups and applications, in the joint meeting of the NZMS, AustMS and AMS (Uni. of Auckland), slides will appear.
2. (Upcoming) November 2024: Combinatorial structures in trace polynomials of function groups (or some such title), at the 8th Australian Algebra Conference in Canberra, slides will appear.
3. 24 July 2024: Is \( \mathrm{PSL}(2,\mathbb{Z}) \) discrete?, in the Topology Seminar (Monash University), slides.
4. 10 May 2023: The dynamic in the static: Manifolds, braids, and classical number theory, in the RePS at Universität Leipzig, slides.
5. 21 September 2022: What is a Kleinian group?, a talk aimed at undergraduates and beginning postgraduate students in the Australian Postgraduate Algebra Colloquium, slides, recording.
6. 4 May 2022: Pictures of hyperbolic spaces, in the Discrete Mathematics and Geometry Seminar (TU Berlin), slides.
7. 17 March 2022: Strange circles: The Riley slice of quasi-Fuchsian space, in Pedram Hekmati's seminar on moduli spaces (Uni. of Auckland), slides.
8. 6 December 2021: The Farey polynomials, for the Groups and Geometry retreat on Waiheke Island, presentation slides.
9. 2 December 2021: The Riley slice, contributed talk for the MATRIX workshop on groups and geometries, presentation slides, recording.
10. 1 April 2021: Real varieties of spherical designs, in the Algebra and Combinatorics Seminar (Uni. of Auckland), presentation slides.

Click for a list of other talks, lecture notes, and other ephemera.

1. November 2024: Variations on a theme of Wielenberg.
2. July 2024: Reading group Hyperbolic Knot Theory, miscellaneous notes.
3. 17, 18 July 2024: Guest lectures for MATHS 782 "Geometric Group Theory" (Uni. of Auckland), lecture notes for lecture 1b (surfaces), lecture notes for lecture 2a (conformal maps), miscellaneous problems, some books
4. July-August 2023: Minicourse on knot theory and geometry (Uni. of Auckland), see below.
5. 16 March 2023: Connectedness of the Hilbert scheme in reading group of Javier and Angel, lecture notes.
6. 17 to 20 January 2023: Apocrypha and ephemera on the boundaries of moduli space, a minicourse at the Uni. of Auckland (also 7 December 2022 at MPI). See below!
7. 10 October 2022: Uniformisation, equivariance, and vanishing: three kinds of functions hanging around your Riemann surface, at MPI, Lecture notes.
8. 3 August 2022: Reproducibility in Computer Algebra (MPI MIS), handouts for practical activity (event co-organised with Christiane Görgen and Lars Kastner).
9. 15 July 2022: On the MathRepo page "Farey Polynomials", in the MathRepo: Data for and from your Research event (MPI MIS), slides.
10. 24 May 2022: Projective varieties over \(\mathbb{C}\), in the Lorentzian polynomials day which I organised, slides.
11. 27 April 2022: Strange circles: The Riley slice of quasi-Fuchsian space, in the Seminar on Nonlinear Algebra (MPI MIS), slides.
12. 19 July 2021: The moulding of hyperbolic clay: Deformation spaces of Kleinian groups (Uni. of Auckland), presentation slides.
13. 8 June 2021: Some properties of \(2 \times 2 \) matrices, in the UoA Dept. of Mathematics Student Research Conference, extended abstract, presentation slides.
    [This talk was one of the four winners of the Best Talk award, along with the talks of Isabelle Steinmann, Chris Pirie, and David Groothuizen Dijkema.]
14. Very rough lecture notes for the graduate seminar I taught on Kleinian groups in Semester 1, 2021: PDF 1, PDF 2, further reading list, and post-mortem.
15. 23 July 2020: Toric varieties (Uni. of Auckland), presentation slides.

Deformation spaces of rank two Kleinian groups

A rank two Kleinian group is a discrete subgroup of \( \mathrm{PSL}(2,\mathbb{C}) \) generated by two elements. If the group is non-elementary, then it is related in complicated and interesting ways to hyperbolic 3-orbifolds that have boundary at infinity consisting of a genus two Riemann surface.

A graph curve is an algebraic curve consisting of a number of thrice-marked spheres, each marked point corresponding to a node (a transverse intersection of two components). Each graph curve of genus two comes from a trivalent graph on two vertices and three edges. There are exactly two such graphs: the theta graph (each edge joins both nodes), and the handcuff graph (one edge joins the nodes, and the others begin and end on the same node). Both of these graphs are homotopy retracts of the genus two handlebody. Therefore there are only two graph curves of genus two.

On the edge of the deformation space of 3-manifolds with genus two surface at infinity there lie manifolds with the same configuration of spheres at infinity: pairs of thrice-punctured spheres with rank one cusps, with incidence graph a trivalent graph with two vertices and three edges (the incidence graph has vertices correspoding to topological components of the surface and edges corresponding to nodes).

By Thurston's ending lamination theorem (proved for this special case by Minsky and Miyachi), on the boundary of the 3-manifold space you get a different limit for each choice of embedding of the trivalent graph into the handlebody, and you can also take limits of such choices to get 'degenerate' orbifolds—the graphs might even be knotted! Conversely every boundary point arises in this way. So there is a very complicated map from the space of these boundary groups (which is basically a Teichmüller space, up to a small quotient) to the space of graph curves (which has two points). It turns out that this complicated map is basically reflecting the geometry of two-bridge links. Manifolds on the boundary that correspond to handcuff graphs arise from two-bridge links with two components, and manifolds corresponding to theta graphs arise from two-bridge knots. The knots do not live inside the deformation spaces, but they lie on tendrils of discrete groups that creep out beyond the moduli spaces.

The Riley slice is the space of Kleinian groups generated by two parabolic elements such that the quotient manifold is a Conway ball: a 3-ball with two arcs drilled out. Choosing a way of arranging these arcs into a rational tangle is equivalent to picking a simple closed curve on the boundary sphere; suppose that this curve is represented by a hyperbolic element \( W_{p/q} \) with trace \( \mathrm{tr}\, W_{p/q} < -2 \) in the holonomy group of the manifold (actually, you need to pick the correct component of the set of points where this word is hyperbolic, but this is immaterial for the time being). The boundary of the deformation space can be reached by smoothly deforming \( W_{p/q} \) until it is parabolic (trace equals \( -2 \)). Keep deforming \( W_{p/q} \) so that its trace decreases; the group is no longer discrete except sporadically, and these discrete groups correspond to replacing the parabolic arc with a cone arc. Eventually you reach \( \mathrm{tr}\, W_{p/q} = 2\), and in fact \( W_{p/q} = 1 \). You have now reached the fundamental group of the \( p/q \) 2-bridge link. The arc (which has now vanished to become a solid part of the knot complement) is an upper or lower unknotting tunnel for the knot; and the point on the boundary of the deformation space where this arc was parabolic corresponds to the manifold where both the knot and the unknotting tunnel have been drilled out as parabolic arcs from \( \mathbb{S}^3 \).

If this sounds interesting:

• You might want to start with our expository article Concrete one complex dimensional moduli spaces of hyperbolic manifolds and orbifolds (joint work with Gaven Martin and Jeroen Schillewaert) which we wrote to give historical and mathematical background: we aimed for this to be accessible to beginning graduate students with only a little complex analysis and topology knowledge.
• (To appear) A comprehensive study of the groups generated by pairs of parabolic and elliptic elements, following work of Keen and Series and various others (joint with Martin and Schillewaert).
• Various authors have written papers in different areas. Some of the most important to us include The Riley slice of Schottky space (Keen and Series), Parabolic representations of knot groups (Riley), Classification of non-free Kleinian groups generated by two parabolic transformations (Akiyoshi, Ohshika, Parker, Sakuma and Yoshida), Cusps in complex boundaries of one-dimensional Teichmüller space (Miyachi), and The tree of knot tunnels (Cho and McCullough). We included a longer (but still not close to comprehensive) list of historical references in the expository article we linked above.
• Work related to the discreteness problem: Approximations of the Riley slice (joint with Martin and Schillewaert); Bounding deformation spaces of 2-generator Kleinian groups (joint with Gong, Martin, and Schillewaert)
• A detailed abstract combinatorial study of certain polynomials which control the geometry of these groups: The combinatorics of Farey words and their traces (joint with Martin and Schillewaert).
• Brief note on the relationships between moduli of Schottky groups and moduli of algebraic curves, see also below.

A zoo of Kleinian groups

There are several useful 'zoos' of Kleinian groups with interesting properties; I collected several interesting groups and families of groups from a few sources, and you can find their limit sets on this page.

These images were produced using the Bella computational package for Kleinian groups. (Bella stood for Better Limit Set Drawer.)

A previous version of this package: Riley slice computational package (GitHub). With this earlier package I produced some animations, and some more limit sets. Some more visually impressive animations can be found on the website of Emily Dumas.

Minicourse on knot theory and geometry

In July 2023 I organised a minicourse on knot theory at the University of Auckland, focusing on the representation theory of holonomy groups. View the abstract or download the latest version of the notes.

Erratum: the Seifert surface of the figure eight knot drawn in the notes is not correct. A correct application of Seifert's algorithm and associated sketch may be found in L. Kauffman, On knots (Princeton), cited at that point in the notes.

There will be eight lectures over four weeks in 303.148 (for the first two weeks at least):

	Wed, 2pm	Fri, 2pm
Classical knot theory	5 Jul: Basics	7 Jul: Fundamental group
Geometric knot theory	12 Jul: Knot complements	14 Jul: Hyperbolic invariants
Braids	19 Jul: Two-bridge knots	21 Jul: Braids and mapping class groups
Knot polynomials	26 Jul: Classical	28 Jul: Quantum

Josh Lehman gave the lecture on mapping class groups and Lavendar Marshall gave the lecture on the Alexander polynomial.

Some useful links:

• From lecture 1.1: Conway: Knots Don't Cancel
• Hyperbolic geometry background for week 2: an intro, VR
• From lecture 2.1: The Geometries of 3 Manifolds, Belt trick, (2,3) Seifert fibration
• Problem session 1 (17 July): problems
• Problem session 2 (31 July): problems

Seminar on deformations of infinite volume hyperbolic 3-folds

Henry Moore: Bronze Form (1988). In situ, Wellington Botanic Garden ki Paekākā. In July 2024 (University of Auckland) and possibly August 2024 (Monash University) I will teach a seminar on the deformation theory of infinite volume hyperbolic 3-manifolds; abstract and background. Notes will appear here. We will cover similar material to Apocrypha and ephemera on the boundaries of moduli space as described below but in much more technical detail. Students who have studied differential geometry and complex analysis are welcome to attend.

Manifesto. We look for a computable, geometrically meaningful coordinate system for the space of representations \( \pi_1(M) \to \mathsf{PSL}(2, \mathbb{C}) \) where \( M \) is a hyperbolic 3-fold. Work of Thurston, Keen and Series, Gilman, Maskit, Kra, Marden, Akiyoshi, Sakuma, Wada, Weeks, Yamashita et cetera gives us hope that such a coordinate system exists for discrete and faithful representations, and that continued deformation of groups along extensions of coordinate lines continues to give meaningful information about indiscrete groups which represent geometrically related cone manifolds.

2023 seminar: Apocrypha and ephemera on the boundaries of moduli space

I taught a minicourse at UoA from 16 17-20 Jan 2023. The goal was to explain the rough structure of the following equivalent objects: (i) the moduli space of Schottky groups; and (ii) the space of hyperbolic 3-manifolds with visual boundary a compact Riemann surface (handlebody). In the process we will learn some of the Birman theory of braid groups, some knot theory, some of the quasi-conformal deformation theory of Kleinian groups, and a lot of geometric topology. The only prerequisite is comfort thinking about quotients in metric spaces and some algebraic topology, but we will go fast and so you should expect to lose track of some details fairly quickly. To try to fix this I will also be handing out problem sheets (which will include some "basic" problems, some research level problems, and some computational problems). Each lecture should also be fairly self-contained. People who attended the graduate seminar/course I taught in Sem 1 of 2021 will find things easier but it is not necessary at all for you to have followed that. The talk in Nelson by Benson Farb is also very good preparation.

I anticipate 5 lectures, at 2PM every day in 303.257 (this schedule is only guaranteed for the first talk, I think some of the more enthusiastic people will want more time to discuss the ideas and so we will wing it as we go). The lectures will be:-

1. A crash course in Kleinian groups problems
2. Sociology problems
3. B-groups and other degeneracies problems
4. Braids, links, and mapping class groups
5. ???

I have written some rough notes which indicate the direction of the conjectured moduli space structure. These notes are not complete.

Representation of algebraic curves by Schottky groups

What is a Riemann surface?

Analytically:

A 2-dimensional manifold (or orbifold) admitting a chart of conformal maps into \( \mathbb{C} \) with conformal transition maps.

Algebraically:

An algebraic curve over \( \mathbb{C} \).

Geometrically:

The quotient of a 2-dimensional geometric manifold by a discrete group of isometries.

There are two kinds of theorems which relate the different viewpoints. First, theorems on rings of functions: the ring of global meromorphic functions on an analytic Riemann surface is isomorphic to a one-dimensional function field over \( \mathbb{C} \) and defines a birationality class of algebraic varieties, setting up an equivalence between the analytic and algebraic worlds. Secondly, uniformisation theorems. Traditionally, one uniformises general (genus \( \geq 2 \) ) analytic Riemann surfaces by Fuchsian groups, i.e. one writes the surface as a quotient \( \mathbb{H}^2/G \) where \( G \) is a discrete group of hyperbolic isometries and is identified with the holonomy group of the surface. However, it is also possible to uniformise all Riemann surfaces by a class of groups of isometries of \( \mathbb{H}^3 \) by considering action on the boundary at infinity.

I survey the three different worlds and some classical theorems (with many examples of Kleinian and Fuchsian groups) in the notes Uniformisation, equivariance, and vanishing—Three kinds of functions hanging around your Riemann surface. In addition, some thoughts on relationships between moduli of Schottky groups and moduli of algebraic curves. Notes on \( G \)-automorphic functions for \( G \) Kleinian. What do Schottky groups look like over more general objects? Schottky groups over \( \mathbb{Q}_p \).

Lorentzian polynomials and algebraic geometry on matroids

If \( X \) is a sufficiently nice variety, the Chow group \( A^*(X) \) provides a homology theory on \( X \); in fact, it admits a ring structure coming from the intersection product. It turns out that such a theory can be made to work on more general spaces, for example one can define a Chow ring for matroids; then the various Hodge-type results (Poincaré duality, the hard Lefschetz theorem, and the Hodge-Riemann relations) carry over. Various nice polynomials can be defined with respect to this generalised Hodge theory and the associated cones of 'ample divisors' (which turn out to be submodular functions); these are the Lorentzian polynomials of Brändén and Huh.

A Day of Geometry and Lorentzian Polynomials

At the end of May 2022 there was a seminar at the Institut Mittag-Leffler on the work of Branden, Huh, Katz, and various other people on Lorentzian polynomials and the geometry of matroids; before this event on Tuesday 24 May, I organised a very informal Zoom workshop on some of the geometric background material.

Abstract. Even if you do not know what Lorentzian polynomials are, you may have heard of Minkowski volume polynomials, the polynomials of the form \( \mathrm{vol}(x_1 K_1 + \cdots + x_n K_n) \) where \( K_1,\ldots,K_n \) are convex bodies—and these are somehow the "canonical examples" of Lorentzian polynomials. The goal of the workshop is to give many different examples of Lorentzian polynomials arising in geometry. The talks will be very informal, non-technical, and have many pictures.

The final schedule was as follows (all times are CET). Many of the speakers have kindly allowed me to share their slides and/or lecture notes.

1. 9.30am—Matroids and chromatic polynomials (Tobias Boege, MPI MiS): Slides
2. 10:15am—Varieties over C and embeddings into projective space via elliptic curves (Lukas Zobernig, The University of Auckland): Slides
3. 11:00am—Hyperbolic polynomials (Hisha Nguyen, V.N. Karazin Kharkiv National University)

• Break (hopefully the morning talks are finished by 11:45, or 12 at the latest if we run over time).

4. 1:30pm—Convex geometry & mixed volumes (Mara Belotti, TU Berlin): Slides
5. 2:15pm—Projective varieties over \( \mathbb{C} \) (Alex Elzenaar, MPI MiS): Slides

Some background material

• Petter Brändén, Jonathan Leake: Lorentzian polynomials on cones and the Heron-Rota-Welsh conjecture
• Petter Brändén, June Huh: Lorentzian polynomials (direct multivariable-calculus proof of the theory without mentioning Hodge theory). See also this lecture by June Huh
• Matthew Baker: Hodge theory in combinatorics (an expository paper)
• Karim Adiprasito, June Huh, Eric Katz: Hodge Theory for Combinatorial Geometries (definition of the Hodge ring for arbitrary matroids)

Spherical designs

A spherical \((3,3)\)-design in \( \mathbb{R}^3 \) of 16 vectors. Spherical \((t,t)\)-designs are arrangements of points on the sphere (possibly with weights) which are spaced 'far apart from each other': they are finite sets in \( \mathbb{R}^d \) such that the integral over the sphere of each homogeneous polynomial of degree \(2t\) in \( d \) variables is equal to its average value on the set. There are generalisations of this definition to subsets of \( \mathbb{C}^d \) and \( \mathbb{H}^d \) (the \(d\)-fold product of the Hamiltonian quaternion algebra, not hyperbolic \(d\)-space!).

Optimal designs and near-designs

Shayne Waldron and I have a paper in preparation: Putatively optimal projective spherical designs with little apparent symmetry, computing various spherical designs in order to find those of minimal order; a large set of designs and near-designs are archived on on Zenodo at DOI: 10.5281/zenodo.6443356. You can look at the code used to generate these on GitHub; it uses the Manopt optimisation toolbox. This work was was funded in part by a University of Auckland Summer Research Scholarship (2019-20). You can view the final report for the scholarship.

Spherical designs and sums of squares

• Talk at the UoA algebra and combinatorics seminar, April 2021: Presentation slides.

BSc(hons) dissertation and MSc thesis

I completed my BSc(Hons) dissertation in 2020 under the supervision of Dr. Jeroen Schillewaert.

• My dissertation: Toric Varieties.
• Half-year postgraduate project presentations, 2020: Presentation slides.

My Master of Science thesis was completed in 2021-22 in the Department of Mathematics at the University of Auckland, under the supervision of Dist. Prof. Gaven Martin (NZ Institute of Advanced Study, Massey University) and Dr. Jeroen Schillewaert. For more information see the section on deformation spaces above.